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EQUILIBRIUM OF FLEXIBLE CORDS                65
EQUILIBRIUM OF FLEXIBLE CORDS
But y = a, when x = 0, therefore c' = 0.    Thus we get
ix
?/ = acos— , ct
, . (9)
= a cosh -,                                   (10)
a
I     X               X\
= -4ei+e~),                         (11)
2 \                />                                           v.    /
/TT     /    "'2;               twaA
~ 2w^6            ''                     ^    ^
which are different forms of the equation of a catenary.
DISCUSSION.—Expanding equation (12) by Maclaurin's Theorem*)* we obtain
y = a  1+"-f~)+"(-J + • • •   •              (13)
In the neighborhood of the lowest point of the cable the value of x is small, therefore in equation (13) we can neglect all the terms which contain powers of x higher than the second. Thus the equation
represents, approximately, the curve in the neighborhood of the lowest point. It will be observed that (14) is the equation of a parabola. This result would be expected since the curve is practically straight in the neighborhood of 0 and consequently the horizontal distribution of mass is very nearly constant, which is the important feature of the Suspension Bridge problem.
The nature of those parts of the curve which are removed from the lowest point may be studied by supposing x to be
X
large. Then since e ~a becomes negligible equation (11) reduces to
V-fA                                  (15)
* Sec Appendix Avm.              t See Appendix Av.